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ON OPTIMAL PROBABILISTIC ALGORITHMS FOR SAT
YIJIA CHEN, J ¨ ORG FLUM, AND MORITZ M ¨ ULLER
1. Introduction. A major aim in the development of algorithms for hard
problems is to decrease the running time. In particular one asks for algorithms
that are optimal: A deterministic algorithm Adeciding a language L⊆Σ∗is
optimal (or (polynomially) optimal or p-optimal) if for any other algorithm B
deciding Lthere is a polynomial psuch that
(1) tA(x)≤p(tB(x) + |x|)
for all x∈Σ∗. Here tA(x) denotes the running time of Aon input x. If (1) is
only required for all x∈L, then Ais said to be an almost optimal algorithm for
L(or to be optimal on positive instances of L).
Various recent papers address the question whether such optimal algorithms
exist for NP-complete or coNP-complete problems (cf. [1]), even though the prob-
lem has already been considered in the seventies when Levin [4] observed that
there exists an optimal algorithm that finds a witness for every satisfiable propo-
sitional formula. Furthermore the relationship between the existence of almost
optimal algorithms for a language Land the existence of “optimal” proof systems
for Lhas been studied [3, 5].
Here we present a result (see Theorem 2.1) that can be interpreted as stating
that (under the assumption of the existence of one-way functions) there is no opti-
mal
probabilistic algorithm for Sat.
2. Probabilistic speed-up. For a propositional formula αwe denote by kαk
the number of literals in it, counting repetitions. Hence, the actual length of any
reasonable encoding of αis polynomially related to kαk.
Theorem 2.1. Assume one-way functions exist. Then for every probabilistic
algorithm Adeciding Sat there exists a probabilistic algorithm Bdeciding Sat
such that for all d∈Nand sufficiently large n∈N
Pr hthere is a satisfiable αwith kαk=nsuch that
Adoes not accept αin at most (tB(α) + kαk)dstepsi≥1
5.
Note that tA(α)and tB(α)are random variables, and the probability is taken over
the coin tosses of Aand Bon α.
1

2 YIJIA CHEN, J ¨ ORG FLUM, AND MORITZ M ¨ ULLER
Here we say that a probabilistic algorithm Adecides Sat if it decides Sat as
a nondeterministic algorithm, that is
α∈Sat =⇒Pr[Aaccepts α]>0,
α /∈Sat =⇒Pr[Aaccepts α] = 0.
In particular, Acan only err on ‘yes’-instances.
Note that in the first condition the error probability is not demanded to be
bounded away from 0, say by a constant ε > 0. As a more usual notion of
probabilistic decision, say Adecides Sat with one-sided error εif
α∈Sat =⇒Pr[Aaccepts α]>1−ε,
α /∈Sat =⇒Pr[Aaccepts α] = 0.
For this concept we get
Corollary 2.2. Assume one-way functions exist and let ε > 0. Then for every
probabilistic algorithm Adeciding Sat with one-sided error εthere exists a prob-
abilistic algorithm Bdeciding Sat with one-sided error εsuch that for all d∈N
and sufficiently large n∈N
Pr hthere is a satisfiable αwith kαk=nsuch that
Adoes not accept αin at most (tB(α) + kαk)dstepsi≥1
5.
This follows from the fact that in the proof of Theorem 2.1 we choose the
algorithm Bin such way that on any input αthe error probability of Bon αis
not worse than the error probability of Aon α.
3. Witnessing failure. The proof of Theorem 2.1 is based on the following
result.
Theorem 3.1. Assume that one-way functions exist. Then there is a probabilis-
tic polynomial time algorithm Csatisfying the following conditions.
(1) On input n∈Nin unary the algorithm Coutputs with probability one a
satisfiable formula βwith kβk=n.
(2) For every d∈Nand every probabilistic algorithm Adeciding Sat and suf-
ficiently large n∈N
Pr Adoes not accept C(n)in ndsteps≥1
3.
In the terminology of fixed-parameter tractability this theorem tells us that
the parameterized construction problem associated with the following parame-
terized decision problem p-CounterExample-Sat is in a suitably defined class
of randomized nonuniform fixed-parameter tractable problems.

ON OPTIMAL PROBABILISTIC ALGORITHMS FOR SAT 3
Instance: An algorithm Adeciding Sat and d, n ∈Nin
unary.
Parameter: kAk+d.
Problem: Does there exist a satisfiable CNF-formula αwith
kαk=nsuch that Adoes not accept αin ndmany
steps?
Note that this problem is a promise problem. We can show:
Theorem 3.2. Assume that one-way functions exist. Then the problem
p-CounterExample-Sat is nonuniformly fixed-parameter tractable.1
This result is an immediate consequence of the following
Theorem 3.3. Assume that one-way functions exist. For every infinite set I⊆N
the problem
SatI
Instance: A CNF-formula αwith kαk ∈ I.
Problem: Is αsatisfiable?
is not in PTIME.
We consider the construction problem associated with the decision problem
p-CounterExample-Sat, that is:
Instance: An algorithm Adeciding Sat and d, n ∈Nin
unary.
Parameter: kAk+d.
Problem: Construct a satisfiable CNF-formula αwith kαk=
nsuch that Adoes not accept αin ndmany steps,
if one exists.
We do not know anything on its (deterministic) complexity; its nonuniform fixed-
parameter tractability would rule out the existence of strongly almost optimal
algorithms for Sat. By definition, an algorithm Adeciding Sat is a strongly
almost optimal algorithm for Sat if there is a polynomial psuch that for any
other algorithm Bdeciding Sat
tA(α)≤p(tB(α) + |α|)
for all α∈Sat. Then the precise statement of the result just mentioned reads
as follows:
1This means, there is a c∈Nsuch that for every algorithm Adeciding Sat and every d∈N
there is an algorithm that decides for every n∈Nwhether (A, d, n) is a positive instance of
p-CounterExample-Sat in time O(nc); here the constant hidden in O( ) may depend on A
and d.

4 YIJIA CHEN, J ¨ ORG FLUM, AND MORITZ M ¨ ULLER
Proposition 3.4. Assume that P6= NP. If the construction problem associated
with p-CounterExample-Sat is nonuniformly fixed-parameter tractable, then
there is no strongly almost optimal algorithms for Sat.
4. Some Proofs. We now show how to use an algorithm Cas in Theorem 3.1
to prove Theorem 2.1.
Proof of Theorem 2.1 from Theorem 3.1: Let Abe an algorithm deciding Sat.
We choose a∈Nsuch that for every n≥2 the running time of the algorithm C
(provided by Theorem 3.1) on input nis bounded by na. We define the algorithm
Bas follows:
B(α) // α∈CNF
1. β←C(kαk).
2. if α=βthen accept and halt.
3. else Simulate Aon α.
Let d∈Nbe arbitrary. Set e:= d·(a+ 2) + 1 and fix a sufficiently large n∈N.
Let Sndenote the range of C(n). Furthermore, let Tn,β,e denote the set of all
strings r∈ {0,1}nethat do not determine a (complete) accepting run of Aon β
that consists in at most nemany steps. Observe that a (random) run of Adoes
not accept βin at most nesteps if and only if Aon βuses Tn,β,e, that is, its first
at most nemany coin tosses on input βare described by some r∈Tn,β,e. Hence
by (2) of Theorem 3.1 we conclude
(2) X
β∈SnPr[β=C(n)] ·Pr
r∈{0,1}ne[r∈Tn,β,e]≥1
3.
Let α∈Snand apply Bto α. If the execution of β←C(kαk) in Line 1
yields β=α, then the overall running time of the algorithm Bis bounded by
On2+tC(n)=O(na+1)≤na+2 for nis sufficiently large. If in such a case a
run of the algorithm Aon input αuses an r∈Tn,α,e, then it does not accept α
in time ne=n(a+2)·d+1 and hence not in time (tB(α) + kαk)d. Therefore,
Pr hthere is a satisfiable αwith kαk=nsuch that
Adoes not accept αin at most (tB(α) + kαk)dstepsi
≥1−Pr for every input α∈Snthe algorithm Bdoes not generate α
in Line 3, or Adoes not use Tn,α,e
= 1 −Y
α∈Sn(1 −Pr[α=C(n)]) + Pr[α=C(n)] ·Pr
r∈{0,1}ne[r /∈Tn,α,e]
= 1 −Y
α∈Sn1−Pr[α=C(n)] ·Pr
r∈{0,1}ne[r∈Tn,α,e]

ON OPTIMAL PROBABILISTIC ALGORITHMS FOR SAT 5
≥1− Pα∈Sn1−Pr[α=C(n)] ·Prr∈{0,1}ne[r∈Tn,α,e]
|Sn|!|Sn|
= 1 −1−Pα∈SnPr[α=C(n)] ·Prr∈{0,1}ne[r∈Tn,α,e]
|Sn||Sn|
≥1−1−1
3· |Sn||Sn|
≥1
5.
Theorem 3.1 immediately follows from the following lemma.
Lemma 4.1. Assume one-way functions exist. Then there is a randomized poly-
nomial time algorithm Hsatisfying the following conditions.
(H1) Given n∈Nin unary the algorithm Hcomputes with probability one a
satisfiable CNF αof size kαk=n.
(H2) For every probabilistic algorithm Adeciding Sat and every d, p ∈Nthere
exists an nA,d,p ∈Nsuch that for all n≥nA,d,p
Pr Aaccepts H(n)in time nd≤1
2+1 np,
where the probability is taken uniformly over all possible outcomes of the
internal coin tosses of the algorithms Aand H.
(H3) The cardinality of the range of (the random variable) H(n)is superpolynomial
in n.
Sketch of proof: We present the construction of the algorithm H. By the as-
sumption that one-way functions exist, we know that there is a pseudorandom
generator (e.g. see [2]), that is, there is an algorithm Gsuch that:
(G1) For every s∈ {0,1}∗the algorithm Gcomputes a string G(s) with
|G(s)|=|s|+ 1 in time polynomial in |s|.
(G2) For every probabilistic polynomial time algorithm D, every p∈N, and all
sufficiently large `∈Nwe have
Pr
s∈{0,1}`D(G(s)) = 1−Pr r∈{0,1}`+1 D(r) = 1≤1
`p
(In the above terms, the probability is also taken over the internal coin
toss of D.)
Let the language Qbe the range of G,
Q:= G(s)s∈ {0,1}∗.
Qis in NP by (G1). Hence, there is a polynomial time reduction Sfrom Qto
Sat, which we can assume to be injective. We choose a constant c∈Nsuch
that kS(r)k ≤ |r|cfor every r∈ {0,1}∗. For every propositional formula βand